Your understanding of the connections to the brake controller is correct.
WRT the "other" (non brake) wire that's in your kit's wiring harness; it's purpose is to provide a recharging current to a storage battery(ies) on the trailer (like a travel trailer's "house" battery). It is to be hooked (via a 30 fuse or breaker) to your truck's battery or alternator. If you are not pulling a trailer that has an installed battery (e.g. a utility or boat trailer), then that wire is unneeded and can be left neatly tied up near the rear of the truck. However, if there's even a slight chance you might pull a travel trailer (or popup) that does have a battery, then this is as good a time as any to hook up that wire to your truck's alternator or battery. Be sure to use a 10 GA wire and a 30 amp fuse or breaker.